Vulnerability Assessment for Cascading Failure in the Highway Trafﬁc System

sustainability

Article Vulnerability Assessment for Cascading Failure in the Highway Trafﬁc System

Han Liu 1 and Jian Wang 2,*

1 School of Transportation Science and Engineering, Harbin Institute of Technology, No. 73, Huanghe River Road, Nangang District, Harbin 150090, Heilongjiang, China; [email protected] 2 School of Management, Harbin Institute of Technology, No. 73, Huanghe River Road, Nangang District, Harbin 150090, Heilongjiang, China * Correspondence: [email protected]; Tel.: +86-152-4666-1633

Received: 4 June 2018; Accepted: 2 July 2018; Published: 5 July 2018

Abstract: Vulnerability assessment is of great significance to highway traffic system. As the widespread cascading failure may cause serious damage, factor identification is necessary to improve the security level of highway system. In this paper, the Dematel method is applied to identify the vulnerability priorities of important factors, and a new assessment method for highway system vulnerability is presented by considering system cascading failure. The trafﬁc allocation and reallocation are then optimized under user equilibrium assignment model in different scenarios. Finally, the methodology is applied to the Heilongjiang highway network and the results shows that vulnerable sections in highway networks can be determined efﬁciently in our proposed process.

Keywords: highway trafﬁc system; vulnerability; cascading failure

1. Introduction In China, the highway traffic system provides a comfortable and speedy transport function to all citizens, trade and industry nationwide. However, incidents, such as vehicle crashes, road construction, lane blockages, and extreme weather, can make parts of the highway traffic system impassable and thus lead to widespread cascading failure. Taking the low-temperature frost and snow disaster in southern China in 2008 as an example, highway traffic was hindered in many provinces and direct losses of up to 159.5 billion yuan and 162 lifelines were resulted [1]. Taking the low-temperature frost and snow disaster in southern China in 2008 as an example, highway traffic was hindered in many provinces and direct losses of up to 159.5 billion yuan and 162 lifelines were resulted. In order to implement suitable policy to reduce the serious loss caused by congestion effects and delays, there is a need for an assessment of the highway traffic system vulnerability. The concept of a vulnerable system is first defined as a system that operates with a “reduced level of security that renders it vulnerable to the cumulative effects of a series of moderate disturbances” [2]. Timmerman regards vulnerability as affected degree on system performance caused by disasters and other incidents [3]. Up to now, the vulnerability concept has been applied to different research fields, however there is not a fully recognized definition for transport system vulnerability. Deep research about transport vulnerability under extreme climate change is far from comprehensive development [4]. Transportation vulnerability is fit for traffic network analysis because it considers not only the probability but also the consequences of disasters. vulnerability can be regarded as an essential indication of the highway traffic system prone to cascading failures. The methods of assessment for the transport system vulnerability can be divided into two categories: software simulation methods and mathematical modeling methods.

Sustainability 2018, 10, 2333; doi:10.3390/su10072333 www.mdpi.com/journal/sustainability Sustainability 2018, 10, 2333 2 of 12

Software simulation methods assess the vulnerability of transport systems by simulating real-life scenarios and what-if scenarios through simulation software. Berdica etc. simulated the different flow-delay sections of Stockholm transport system and analyzed the main evaluation indicators including the average travel time, speed changes and total travel costs [5]. Lin used graph theory to establish rules of vulnerability system, and simulated vulnerability behavior [6]. Song proposed a new method by embedding the Delphi survey into geographic information system, and assessed regional ecological vulnerability using Monte Carlo simulation [7]. Guardo proposed a software system that can preload Environmental and pesticide data to evaluate potential groundwater vulnerability to pesticides [8]. Ferber simulated different attack strategies by removing different nodes and analyzed the consequence severity of these strategies [9]. Software simulation methods generally consider a particular case. Longer time is needed for simulation under more conditions. Mathematical modeling methods assess the vulnerability and identify the vulnerable sections of transport systems by constructing mathematical models. Asadabadi formulated an investment optimization framework by considering uncertain climate impacts and studied different climate change impacts on roadway networks [10]. Gabriele Landucci developed an analytical approach for risk assessment in transportation system [11]. Matisziw constructed an objective programming model to analyze the potential traffic flow change between the start and end point and identify important sections of the system [12]. A complex transportation system vulnerability model is established by combining fuzzy analytic hierarchy process and the maximum entropy principle [13]. Han developed a new hypercube queuing model to evaluate the vulnerability of urban fire emergency system [14]. Zhang explored the chain rear-end accident mechanism of freeway, and established vulnerability structural models [15]. Yang combined hierarchy analytic process with fuzzy comprehensive evaluation theory, and selected the single vulnerability evaluation index [16]. In the problem of complex traffic system vulnerability assessment, mathematical modeling methods have unique advantages for the accurate and logical characteristics. However, the modeling or solving process is difficult as well as the reasonable assumptions of the vulnerability assessment model. Despite all the efforts in the above areas, no previous study on the highway vulnerability assessment has captured the system cascading failure factors based on general method like Dematel method. Specially, the relationship and interaction of heterogeneous factors causing highway cascading failures have generally been ignored in the vulnerability literature. Separate consideration of the factors may result in inaccurate estimations. Therefore, in this study we using Dematel method to assess the vulnerability of system cascading failure, and then presents a user equilibrium assignment model to optimize the traffic allocation and reallocation. The remainder of this paper is organized as follows. The vulnerability of highway traffic system are analyzed on in Section2. Vulnerability assessment system is established based on our proposed assessment method in Section3. Then the algorithm design under user equilibrium assignment model is illustrated in Section4. A case study of Heilongjiang highway network vulnerability assessment is applied in Section5. Finally the conclusion is drawn in Section6.

2. Analysis on Highway Trafﬁc System Vulnerability

2.1. Vulnerability Factor Structure Highway transportation system is a complex system which is open, non-linear, and self-adaptive. The subsystem of highway traffic system could be divided into four types (shown in Figure1): Road, Vehicle, Human and Environment. Vulnerability factors of the first two subsystems are essential factors, and others are classified into external factors. Sustainability 2018, 9, x FOR PEER REVIEW 3 of 12 Sustainability 2018, 10, 2333 3 of 12

Network topology

Road Road capacity Essential factors Pavement quality

Vehicle Traffic volume

Road maintenance Highway system vulnerability Deliberate destruction Human Management measures External Traffic accidents factors

Geological disasters Environment Weather conditions

FigureFigure 1. Analysis 1. Analysis of of highway highway traffic trafﬁc systemsystem vulnerability. vulnerability.

2.2. Identification Identification and Analysis of VulnerabilityVulnerability Factors by DEMATEL MethodMethod The factorsfactors in in a complexa complex system system may bemay mutually be mutually related inrelated direct orin indirectdirect or ways. indirect As interferences ways. As withinterferences one of the with factor one of may the affects factor themay others, affectssometimes the others, itsometimes is difficult it tois difficult find priorities to find forpriorities factor forimportance. factor importance. Decision-makers Decision-makers may fail may in obtaining fail in obtaining a specific a objectivespecific objective if the disruptions if the disruptions caused causedby the restby the of therest systemof the system cannot cannot be avoided. be avoided. The DEMATEL The DEMATEL method, method, proposed proposed by Fontela by Fontela and andGabus Gabus from from the the Battelle Battelle Memorial Memorial Institute Institute of of Geneva, Geneva, can can be be used used to to find find prioritiespriorities ofof various factors in an interdependentinterdependent system [[17].17]. Factor Factorss of of the the highway highway traffic traffic system are the critical subsystem of a complex highway system, and mutually related, directly or indirectly. Its collapse will seriously interfere otherother sections.sections. Therefore, Therefore, the the identification identification and analysis of vulnerability factors is essential to improve the highway traffi trafficc safety. The DEMATEL method studies the mutual influenceinfluence relationshiprelationship among among multiple multiple factors factors from from the the perspective perspective of graph of graph theory, theory, analyzes analyzes the logical the logicalrelationship relationship among theamong system the influencing system influencing factors, and factors, derives and the derives direct influencethe direct matrix influence through matrix the throughempirical the discrimination empirical discrimination method. Itis method. a method It is of a analyzing method of the analyzing influencing the influencing factors of the factors system. of Thethe system. method The performs method matrix performs operations matrix on theoperations direct influence on the direct matrix, influence eliminates matrix, the mutual eliminates influence the mutualbetween influence them, and between obtains them, the degree and obtains of influence the degr ofee various of influence factors of on various the system. factors The on the purpose system. of Theapplying purpose the of DEMATEL applying the method DEMATEL is to analyze method components is to analyze structure components of vulnerability structure offactors vulnerability which factorsincludes which essential includes factors andessential external factors factors and to avoidexternal the “overfitting”factors to avoid for decision-making. the “overfitting” for decision-making.The steps of the DEMATEL method are described as follows [18]: The steps of the DEMATEL method are described as follows [18]: Step 1: Calculate the average matrix. Step 1: Calculate the average matrix. Step 2: Calculate the initial direct influence matrix. Step 2: Calculate the initial direct influence matrix. Step 3: Derive the full direct/indirect influence matrix. Step 3: Derive the full direct/indirect influence matrix. StepStep 4:4: Set threshold value and obtain the impact-digraph-map. According the steps of the DEMATEL method, the factor identification identification processes are conducted as follows. (i) Determining the significantsignificant factors of thethe system.system. By doing literature review, information collection and field field research on the expressway traffictraffic system, we initially determine the possible vulnerability factor of the system and establish a system ofof possiblepossible vulnerabilityvulnerability factorfactor sets.sets. (ii) Establishing relationship matrix. We We de designsign surveys surveys and and summarize the results before establishing an initial direct impact relationship matrix. Expert experience method is used to judge Sustainability 2018, 9, x FOR PEER REVIEW 4 of 12 Sustainability 2018, 10, 2333 4 of 12 the relationship between any two influencing factors. The influence of one influencing factor on theitself relationship is zero. between any two influencing factors. The influence of one influencing factor on itself is zero.(iii) Calculating the indirect influence relationship matrix. Normalizing the directly influence matrix(iii) to Calculating ensure the influence the indirect degree influence of any relationship two factors matrix.is within Normalizing the range [0, the 1). directlyUsing the influence matrix matrixconversion to ensure method the to influence calculate degree the comprehensive of any two factors impact is matrix, within the the indirect range [0, influence 1). Using relationship the matrix conversionof each factor method can be to obtained. calculate the comprehensive impact matrix, the indirect influence relationship of each(iv) factor Drawing can bea causality obtained. diagram. We take the centrality of the influencing factor as the abscissa and (iv)the Drawingcause of athe causality influencing diagram. factor We as take the the ordinate centrality to analyze of the influencing the influencing factor asfactors the abscissa of the andsystem. the causeIn this of way, the influencing we obtained factor the asimportance the ordinate of toevery analyze vulnerability the influencing factor factors of highway of the system. traffic Insystem this way, and wethe obtained diagram the of importancerelationship of between every vulnerability centrality factordegree of and highway reason traffic degree, system which and are the diagramshown in of Table relationship 1 and Figure between 2 respectively. centrality degree and reason degree, which are shown in Table1 and Figure2 respectively. Table 1. Importance of every vulnerability factors of highway traffic system. Table 1. Importance of every vulnerability factors of highway traffic+ system.− No. Vulnerability Factors Ri Ci RCii RCii

No.S1 VulnerabilityRoad Maintenance Factors R 1.0413i 0.3924Ci Ri1.4337+ Ci Ri0.649− Ci S2 Deliberate Destruction 1.1976 0.7411 1.9387 0.4564 S1 Road Maintenance 1.0413 0.3924 1.4337 0.649 S23 DeliberateTraffic DestructionAccidents 1.19761.4433 0.74111.4016 1.93872.845 0.45640.0417 S34 ManagementTrafﬁc Accidents Measures 1.44330.9232 1.4016 1.3564 2.845 2.2796 0.0417−0.433 S − S45 ManagementTraffic Volume Measures 0.9232 0.8302 1.35642.3165 2.27963.1467 −0.4331.486 S5 Trafﬁc Volume 0.8302 2.3165 3.1467 −1.486 S6 Network Topology 0.3135 1.4164 1.7299 −1.103 S6 Network Topology 0.3135 1.4164 1.7299 −1.103 S7 Pavement Quality 0.4283 0.7269 1.1552 −0.299 S7 Pavement Quality 0.4283 0.7269 1.1552 −0.299 S8 RoadRoad Capacity Capacity 0.8315 0.8315 2.1712.171 3.00253.0025 −−1.3391.339 S9 Weather Weather Conditions Conditions 1.8681 1.8681 0.06260.0626 1.93071.9307 1.80561.8056 S10 Geological Disasters 1.7399 0.0322 1.7721 1.7077 S10 Geological Disasters 1.7399 0.0322 1.7721 1.7077

Figure 2. Relationship between centrality degree and reason degree. Figure 2. Relationship between centrality degree and reason degree.

According to the results by DEMATEL method, the combined effect of the matrix, Table 1and FigureAccording 2 were analyzed: to the results by DEMATEL method, the combined effect of the matrix, Table1and Figure2 were analyzed: (1) Essential factors (1) Essential factors Essential factors which impact on the highway transportation system vulnerability were orderedEssential by descending factors which importance impact as on follow: the highway traffic volume, transportation road capacity, system vulnerabilitynetwork topology, were orderedmanagement by descending measures, pavement importance quality. as follow: We regard trafﬁc these volume, factors road as direct capacity, reasons network which topology, lead to managementvulnerability excitation. measures, pavement quality. We regard these factors as direct reasons which lead to vulnerabilityAmong these excitation. results, the centrality degree of S5 (traffic volume factor) and S8 (road capacity factor) is respectively 3.1467 and 3.0025, which indicates the degree of their influence is important in the system. In the course of the actual operation of the road network, traffic volume and road Sustainability 2018, 10, 2333 5 of 12

Among these results, the centrality degree of S5 (traffic volume factor) and S8 (road capacity factor) is respectively 3.1467 and 3.0025, which indicates the degree of their influence is important in the system. In the course of the actual operation of the road network, traffic volume and road capacity are equivalent to traffic demand and supply for transport. Interactions of both factors determine whether the vulnerability can be stimulated. The centrality degree of S6 (network topology factor) is 1.7229 and its reason degree is −1.103 degrees, which means that the importance of S6 cannot be ignored. Once the network topology of the system is changed, the internal system traffic redistribution would generated, which leads to traffic volume disturbance and vulnerability excitation (2) External factors External factors were also ordered by descending importance as follow: geological disasters, weather conditions, road maintenance, deliberate destruction, traffic accidents. The reason degree of these factors is greater than 0, indicating that influence on the system from external factors is greater than the influence on external factors from the system. External factors are the direct external causes of the system vulnerability. Vulnerability of these five factors is analyzed. The reason degrees of S10 (Geological disasters factor) is 1.7077, the most significant importance of external factors, which is consistent with our perception. Once the geological disaster occurs, it will have a serious impact on the highway system. However, geological disasters have lower frequencies and some of them are predictable and preventable, therefore S10 is not considered as vulnerability influencing factor in the conventional highway network design, management and operation. The centrality degree and reason degree of S9 (weather conditions) and S1 (Road maintenance) is not higher than other external factors except S10, which means that they are main vulnerability factors of highway traffic system. The reason degree of S2 (deliberate destruction) is not relatively high, indicating deliberate destruction rarely happen in China. Therefore S2 is not regarded as the main vulnerability factor. As for S3 (traffic accidents), although the reason degree is not high, its centrality degree is 2.845, which is greatest among all of the external factors. Therefore, we regard it as the main vulnerability factor. To sum up, the main vulnerability factors are S9 (weather conditions), S1 (Road maintenance) and S3 (traffic accidents).

3. Vulnerability Assessment System Establishment

3.1. Vulnerability Assessment System Establishment In order to assess the vulnerability of highway transport system based on cascading failure phenomenon, we must identify the status of vulnerable sections. If the traffic volume is less than or equal to the maximum highway capacity, the section is in a normal status; On the contrary, if the traffic flow is more than the maximum highway capacity, the section is in a failed status. In the latter status, traffic flow is reallocated, and an alternative route is chosen by users who have selected the vulnerable section of highway transport system. If the traffic flow is more than the maximum highway capacity after flow redistribution, the other sections would convert from a normal status to a failed status, which causes more serious congestion. As a result, another reallocation of traffic flow is necessary. The cascading failure would not end out until the traffic flow is less than the maximum highway capacity. Traffic flow reallocation is a repeated iteration step which is critical to vulnerability assessment of highway transport system. User equilibrium assignment model (UE model) and system optimal allocation model (SUE model) are widely used in traffic flow reallocation. When the vulnerable section is exposed, users who have selected the vulnerable section need to choose an alternative route. Note that the users cannot fully understand real-time traffic information and other travelers’ choice before making decisions, and they seek to select the optimal path to minimize their transportation cost. Therefore, the UE model can be used to evaluate methods of vulnerability. Sustainability 2018, 10, 2333 6 of 12

The equivalent mathematical formulation is

Z va min z(x) = ∑ ta(va)d(va) a∈A 0

 ij  ∑ f = qij, ∀i, j  k  k  ( ) = ij ij va xa, ya ∑ ∑ ∑ fk δa,k s.t. i j k (1)  ij  f ≥ 0, ∀i, j  k 

ij where, fk is the number of trips between the O-D pair (i, j) that uses path k, ta is the cost of flow v ij using link a, va is the flow in link a, and δa,k equal to one if part of path k between i and j belongs to link a and zero otherwise. Some measure indices are frequently used in complex network assessment. They include point degree, degree distribution, average distance, shortest distance, effectiveness, clustering coefficients, betweenness and so on. As static indices of complex systems, they are often used to compare the effectiveness of different networks. However, the vulnerability excitation of highway network system is a dynamic, concatenate and time-delay process, the static indices cannot accurately measure the vulnerability excitation impact to users in the system. Therefore, we consider the congestion-caused failure process and attempt to define vulnerability measure indices of highway network system. Cascading failure of highway network system caused by vulnerable sections may bring impact to users in two ways. If users at vulnerable sections cannot find an alternative route to reach the destination, their travel demands could not be met, which will result in delays in travel. For another, if users at vulnerable sections reach the destination through a detour on a trip, they would spend more travel time and travel costs to meet their travel demand. In case of vulnerable section failure, if the demand of travelling to destination could not be met, the users have to wait until the vulnerable section becomes normal, or exit the highway and enter other low grade road. Then the traffic supply of the section drops to 0 due to the vulnerable section failure. In this issue, assessment formula proposed by Jenelius is used for failure section [19].

k ∑ ∑ fij k i j Iij = (2) ∑ ∑ fij i j

If the users have to detour to meet their travel demands, the economic costs of travel can be measured by the total cost of travel, see Equation (3). And the travel time cost can be measured by total travel time in highway network, see Equation (4).

0 Ei = ∑ La × (m1 + m2) × va (3) a∈(L−i) where, Ei is the total travel cost after trafﬁc ﬂow allocation due to section failure, La is the Length of section a, m1 is the toll station costs per kilometer, m2 is the fuel costs per kilometer. According to the highway toll station fees standard of Heilongjiang Province, m1 is set to be 0.5 Yuan. m2 is set to be 0.6 Yuan due to the average fuel consumption of vehicles.

0 0 Ti = ∑ vata(va) (4) a∈(L−i) Sustainability 2018, 10, 2333 7 of 12

where, Ti is the total travel time after traffic flow allocation due to section failure, (L − i) represents the scenario that section i is in failure status, va is the traffic flow allocated for section a before iterations, 0 0 va is the traffic flow allocated for section a after iterations, ta(va ) is the section travel time, and we use β BPR function ta(va) = t0[1 + α(va/C) ], where we set 0.15 for α, and 4 for β. The total cost of travel Ci is Ci = δ × Ti + Ei (5) where, δ is time value per hour, and we set 20 for it here. Considering these two factors, we defined vulnerability value Rij as follow,

C R = Ik + (I − Ik ) i (6) ij ij ij C where, C is the total cost of travel without UE equilibrium process. After dimensionless, the vulnerability assessment indices of highway network system are represented as a value between 0 and 1, the formula as follows:

0 Ri − Rmin Ri = (7) Rmax − Rmin

0 where, Ri is the dimensionless vulnerability value, Ri is the original vulnerability value, Rmax is the maximal vulnerability value and Rmin is the minimal vulnerability value for each highway section. The levels of vulnerability are graded as shown in Table2.

Table 2. Vulnerability Index Ranking.

Vulnerability Level Vulnerability Index Ranges High Level [0.75–1.0] Medium Level [0.5–0.75) Low Level [0.25–0.5) No Vulnerability [0–0.25)

3.2. Vulnerability Assessment Process We firstly collect information (including highway network length, number of lanes, the node population of the existing road network, mark the numbers of nodes and links newly created. Then we generate the network topology of the actual system through link information statistics. By applying the UE traffic assignment model to traffic flow reallocation, we can obtain the initial traffic flow. The impedance of failure sections is set to ∞, and the number of iterations k is set to 1. Before each time traffic flow is reallocated, we judge whether the OD traffic demand is met. If so, a separate record of this traffic demand is made, and the traffic demand of the OD matrix is changed to 0. If not, the traffic flow is directly reallocated. The next step is OD traffic matrix redistribution. Through that we judge whether there is a section in which the traffic flow is more than its traffic capacity. The impedance of this section is set to ∞, and the number of iterations k is added by 1. Then the traffic flow based on the new impedance matrix is reallocated. Repeat the step until the traffic flow is less than its traffic capacity in sections whose impedance is not ∞. As soon as the congestion propagation does not continue, effects iterations come to the end. Figure3 shows the algorithm flow. Sustainability 2018, 10, 2333 8 of 12 Sustainability 2018, 9, x FOR PEER REVIEW 8 of 12

FigureFigure 3. 3.SystemSystem vulnerability vulnerabilityassessment assessment ﬂowchart. flowchart.

4.4. Algorithm DesignDesign AccordingAccording to vulnerability vulnerability assessment assessment indices indices and and process process of ofhighway highway network network system, system, we weprogrammed programmed algorithm algorithm for for our our model model by by Matlab Matlab so software.ftware. UE UE equilibrium equilibrium model model was was solved by usingusing thethe steepeststeepest descent descent method, method, in in which which the the basic basic step step is tois ﬁndto find the descentthe descent direction direction and theand step the sizestep [20size]. According[20]. According to the to basic the principles basic principles and steps and of steps the steepest of the steepest descent method,descent solvingmethod, methods solving formethods UE balanced for UE distributionbalanced distribution model are model shown are in Figureshown4 in. Figure 4. TheThe basicbasic formulaformula forfor thethe steepeststeepest descentdescent methodmethod isis asas shownshown inin EquationEquation (8).(8). nn+1 =+λ n − n xnx+1 = xxxn + λ((yyna− xna)) (8)(8) a a

n xn λ where,where, x is is the the trafﬁc traffic ﬂow flow vector vector of of the the n-th n-th iteration. iteration.λ is the is stepthe step size size to be to determined. be determined. Solving equation for iteration step λλis as follow: Solving equation for iteration step is as follow: (yn − xn)tn[xn + λ(yn − xn)] = 0 (9) ∑ nna nna a na a na a()−+−=λ ( ) 0  yyaaxtaa x a x a (9) a  Sustainability 2018, 9, x FOR PEER REVIEW 9 of 12

Sustainability 2018, 10, 2333 9 of 12 Sustainability 2018, 9, x FOR PEER REVIEW 9 of 12

Figure 4. Solving flowchart for user equilibrium allocation model.

5. Case Study FigureFigure 4. Solving 4. Solving flowchart ﬂowchart for for user equilibriumequilibrium allocation allocation model. model.

5.We Case test Study our methodology on a transportation network of Heilongjiang Province, China (see Figure 5). Heilongjiang locates in the northeastern part of the country and contains China’s We testtest ourour methodologymethodology onon aa transportationtransportation networknetwork of HeilongjiangHeilongjiang Province, China (see northernmost point and easternmost point. Its network consists in a total of 19 links and 17 nodes. Figure 5).5). Heilongjiang locates inin thethe northeasternnortheastern partpart ofof thethe countrycountry andand containscontains China’sChina’s Every node can be the origin or the destination. Node No. and City Information are shown in Table northernmost point and easternmost point. point. Its Its networ networkk consists consists in in a a total total of of 19 19 links links and and 17 17 nodes. nodes. 3. Every node cancan bebe thethe originorigin oror thethe destination.destination. Node Node No. No. and and City City Information Information are are shown shown in in Table Table3. 3.

Figure 5. Highway network topology of Heilongjiang Province. FigureFigure 5. Highway 5. Highway network network topology topology of of Heilongjiang Heilongjiang Province. Province.

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Table 3. Node No. and City Information.

Node Node Population Node Population Node Population Node Name Node Name No. Name (Thousands) No. (Thousands) No. (Thousands) 1 Nenjiang 504 7 Harbin 9935 13 Jixi 1859 2 Qiqihaer 5591 8 Hegang 1085 14 Mudanjiang 2528 3 Daqing 2817 9 Jiamusi 2214 15 Suifenhe 68 4 Heihe 1224 10 Shuangyashan 1504 16 Beian — 5 Yichun 1241 11 Tongjiang 180 17 Yilan — 6 Suihua 5770 12 Qitaihe 924

To investigate the trafﬁc volume of Heilongjiang Provincial highway is difﬁcult, thus we use unconstrained gravity model to analysis the OD matrix which is shown in Table4.

P P = i j qij α 2 (10) dij where, qij is the trafﬁc ﬂow from i to j, Pi, Pj are population of i and j.

Table 4. Trafﬁc Demand Matrix.

No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 0 29 6 0 1 5 11 0 1 0 0 0 1 1 0 2 29 0 466 6 12 139 405 8 18 10 1 6 10 20 0 3 6 466 0 4 10 183 803 6 15 8 1 5 7 17 0 4 0 6 4 0 4 19 24 1 2 1 0 1 1 2 0 5 1 12 10 4 0 88 2 4 4 2 0 1 2 4 0 6 5 139 183 19 88 0 4593 16 0 19 1 13 18 44 1 7 11 405 803 24 2 4593 0 41 102 48 3 33 42 116 2 8 0 8 6 1 4 16 41 0 980 80 2 19 10 7 0 9 1 18 15 2 4 0 102 980 0 382 4 62 13 16 0 10 0 10 8 1 2 19 48 80 382 0 6 18 11 8 0 11 0 1 1 0 0 1 3 2 4 6 0 1 1 1 0 12 0 6 5 1 1 13 33 19 62 18 1 0 36 13 0 13 1 10 7 1 2 18 42 10 13 11 1 36 0 118 1 14 1 20 17 2 4 44 116 7 16 8 1 13 118 0 5 15 0 0 0 0 0 1 2 0 0 0 0 0 1 5 0

We used Matlab software to program and run the algorithm of vulnerability assessment. The result is shown in Table5.

Table 5. Result of Section Vulnerability Value.

Section No. 1 2 3 4 5 6 7 i − j 1–2 2–3 3–7 4–16 5–6 5–16 6–7 Rij 1.116 2.426 2.692 1.142 1.933 1.001 3.029 Section No. 8 9 10 11 12 13 14 i − j 6–16 7–14 7–17 8–9 9–10 9–12 9–17 Rij 1.132 1.621 1.312 1.930 1.296 1.356 1.673 Section No. 15 16 17 18 19 20 i − j 10–11 11–13 12–13 12–17 13–14 14–15 Rij 1.040 1.005 1.210 1.129 1.527 1.020

In the assessment S3 (trafﬁc accident factor) is chosen for its main vulnerability character. Assuming that the weather is in normal conditions, highway closure is caused by an accident, which leads to cascading failures. According to Formula (9), the results are shown in Table6. Sustainability 2018, 10, 2333 11 of 12

Table 6. Dimensionless Result of Section Vulnerability Value.

Section No. 7 3 2 5 11 14 9 19 13 10 0 Rij 1.00 0.83 0.70 0.46 0.46 0.33 0.31 0.26 0.18 0.15 Section No. 12 17 4 8 18 1 15 20 16 6 0 Rij 0.15 0.10 0.07 0.06 0.06 0.06 0.02 0.01 0.00 0.00

Table7 shows that Section 7 and 8 are ranked in High Vulnerability Level. In other words, once the vulnerable section in High Level suffered from incidents including vehicle crashes and severe weather, cascading failures probably happen. Section 2 is ranked in Medium Level, and its vulnerability is less serious than Section 7 and 8. Section 5, 11, 14, 9 and 19 have the least extent of vulnerability. The remaining sections are not attributed to the three vulnerability levels above, which means that incidents have little effect to these sections. The results demonstrate that our method is effective to determine vulnerable sections in highway networks.

Table 7. Section Vulnerability Level Classiﬁcation.

Vulnerability Level Section No. High Level 7, 3 Medium Level 2 Low Level 5, 11, 14, 9, 19 None 13, 10, 12, 17, 4, 8, 18, 1, 15, 20, 16, 6

6. Conclusions This paper by identifies the importance priority of highway transportation system vulnerability factors by applying Dematel method. A new assessment method for highway vulnerability is presented under system cascading failure. We establish vulnerability assessment system by considering different situations and UE equilibrium, and propose the location and reallocation model based on vulnerability assessment process. Finally, this methodology is applied to Heilongjiang highway network and the results demonstrate that the method is effective for the vulnerable sections identification in highway networks. Due to the limitation of insufficient data, for the evaluation index of brittle sources, this paper only considered the consequences of failure, but did not include the probability of brittle failure. In future studies, the assessment of brittle sources can be performed by considering these two effects.

Author Contributions: Prof. Jian Wang conceived and designed the experiments; Han Liu performed the experiments, analyzed the data. Both of the authors wrote the paper. Funding: National Natural Science Foundation of China (51578199). Theoretical Research on Traffic Network Design under the Background of Auto-Driven Cloud Plan. Conflicts of Interest: The authors declare no conflict of interest.

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